This disclosure relates generally to methods and systems for analyzing three dimensional digital volumes of material samples to determine properties of the sampled material.
Knowledge of the material properties, also referred to as physical or petrophysical properties, of subsurface rock formations is important for assessing hydrocarbon reservoirs in the earth, and for formulating a development strategy regarding those reservoirs. Traditionally, samples of the rock formation of interest are subjected to physical laboratory tests to determine these material properties. These tests, however, are typically time consuming and expensive. Hence, there is a desire to develop technologies that can obtain reliable estimates of material properties of the subsurface rock, at a fraction of the time and cost of traditional laboratory based approaches.
Direct numerical simulation of material properties from digital images of rock is one promising technology aimed at achieving this objective. To determine the material properties utilizing this approach, an x-ray tomographic image is taken of a rock sample, and a computational experiment is applied on the digital image volume to simulate a specific physical experiment. Material properties such as porosity, absolute permeability, relative permeability, formation factor, elastic moduli, and the like can be determined using this conventional approach.
Direct numerical simulation has the potential to provide material properties of difficult rock types, such as tight gas sands or carbonates, within a timeframe that is substantially shorter than that required for experimentally derived material properties. This is because the process for achieving the physical conditions necessary for a specific experiment, such as full water saturation, to proceed can be quite slow. In contrast, the analogous numerical conditions that replicate the physical experiment are readily and rapidly achievable.
For most rock types, it is necessary to acquire high resolution images of the rock to resolve its pore space. This usually requires the images to be taken on a small rock sample, for example a sample extracted from a larger rock sample such as a plug, rotary core or whole core. However, pore system heterogeneity may not always be well-represented within such a small imaged portion of the rock. In some cases, the computational domain is too small for the pore system and the computed material properties fluctuate significantly about the true value for the rock.
This issue is often ignored in conventional direct numerical simulation of material properties from experimentally acquired images. Rather, computations are performed on the largest possible volume extractable from the image, without regard to whether the computational domain is appropriate for the pore system. Thus, the computed material properties may be in error due to lack of pore system representativeness.
To establish whether computed material properties are impacted by a lack of pore system representativeness, Representative Elementary Volume (REV) analysis is sometimes performed. This approach is quantitative, in that if a representative elementary volume is shown to exist, its size is also determined. By conducting this analysis, the effect of pore scale variability and scale dependence on material properties can be directly assessed.
Traditionally, the REV has been defined as the volumetric extent of a rock from which computational experiments or physical measurements will return values that are representative of the larger, or macroscopic, homogeneous rock mass. That is, the REV is defined as the sample volume size at which the physical parameter being computed or measured from the sample volume is not dependent on the particular location of the sample volume within the overall mass. Conversely, the data from computational measurements or experiments made on a computational domain or rock sample of a volume smaller than the REV may not accurately represent the pore system of the rock mass macroscopically, but the physical parameter being computed or measured will vary depending on the location of the computational domain within the rock mass. As the size of the sample volume approaches that of the REV, the computed or measured parameter will tend toward a true representative value. Computations and experiments performed on volume sizes greater than the representative volume will return values equivalent to those obtained on the volume defined as the REV (i.e., the representative value), provided that no macroscale heterogeneities are present.
FIG. 1 illustrates the traditional definition of the REV for porosity of a porous medium. In FIG. 1, the sample volume is denoted by ΔVi, the REV volume is denoted by ΔV0, and ni represents the void space volume divided by the volume of the sample. In sampling volumes ΔVi<ΔV0, only a small number of pores and grains are present. This situation is shown in the left-hand pane of FIG. 2, in which sample volumes ΔVi are smaller than the REV ΔV0, and do not include a sufficient number of pores and grains to permit a physically meaningful statistical average of porosity to be determined. As a result, the porosity calculation over these sample volumes will tend to reflect local pore scale variability rather than accurately represent the porosity of the overall porous medium. As the sample volume size decreases further below the REV, the calculated ratio of void space to total volume will approach one or zero, depending on whether the centroid P of the sample volume happens to be situated within a pore or a grain. In that case, the value of ni is dominated by local micro scale variability of the pore space.
On the other hand, sample volumes ΔVi of a size at or above the REV ΔV0 contain a sufficient number of pores and grains to permit a physically meaningful statistical average of the overall rock to be determined from a sample. This is shown in the right-hand pane of FIG. 2, in which sampling volumes ΔVi are greater than the REV ΔV0, such that porosity calculation for volume will reflect the actual porosity value of the porous medium (i.e., the relative pore space ni=φ). For sample volumes ΔVi>>ΔV0 of a homogenous porous medium, the calculated or measured porosity is essentially constant at the same porosity as represented at the REV sample volume size. However, for an inhomogeneous porous medium, macroscale inhomogeneities will cause fluctuations in the porosity, even over a population of sample volumes ΔVi>>ΔV0.
This classical definition of the REV underpins the continuum framework for definition of material properties of porous materials. That is, porosity, permeability, formation factor, etc. are all defined as volumetric averages of microscopic properties at the REV volume. However, an REV for one material property, such as porosity, may not necessarily be the REV for another material property, such as permeability.